# Gradient (or slope) of a Line, and Inclination

**Application:** Road sign, indicating a steep gradient.

A `15%` road gradient is equivalent to `m = 0.15`.

The **gradient** (also known as **slope**) of a line is defined as

`"gradient"= text(vertical rise)/text(horizontal run`

In the following triangle, the gradient of the line is given by: `a/b`

In general, for the line joining the points (*x*_{1}, *y*_{1}) and (*x*_{2},
*y*_{2}), we have:

We can now write the fomula for the slope of a line.

## Gradient of a Line Formula

We see from the diagram above, that the **gradient** (usually written *m*) is given by:

`m=(y_2-y_1)/(x_2-x_1`

## Interactive graph - slope of a line

You can explore the concept of slope of a line in the following JSXGraph (it's not a fixed image).

**Drag** either point A or point B to investigate how the gradient formula works. The numbers will update as you interact with the graph.

Notice what happens to the sign (plus or minus) of the slope when point B is above or below A.

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

Sometimes the explanation boxes overlap. It can't be helped!

If you get lost, you can always refresh the page.

### Example

Find the slope of the line joining the points (-4, -1) and (2, -5).

### Positive and Negative Slopes

In general, a **positive slope **indicates the
value of the dependent variable **increases** as we go left to
right:

[The **dependent variable** (usually *x*) in the above graph is the *y*-value.]

A **negative slope** means that the value of the dependent variable (usually *y*) is **decreasing** as we go left to right:

## Inclination

We have a line with slope *m* and the angle that the line makes with the
*x*-axis is α.

From trigonometry, we recall that the tan of angle α is given by:

`tan\ alpha=text(opposite)/text(adjacent)`

Now, since slope is also defined as opposite/adjacent, we have:

This gives us the result:

tan

α=m

Then we can find angle *α* using

α= arctanm(That is,

α= tan^{-1 }m)

This angle α is called the **inclination** of the line.

### Exercise 1

Find the inclination of the line with slope `2`.

**NOTE:** The size of angle *α* is (by definition) only
between `0°` and `180°`.

### Exercise 2

Find the slope of the line with inclination *α* = 137°.

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